Integral closure in an infinite algebraic extension

Question

If $A$ is a principal ideal domain and $L/Q(A)$ a finite field extension, then it follows from Krull-Akizuki theorem that the integral closure of $A$ in $L$ is a Dedekind domain. Now if $L/Q(A)$ is an infinite algebraic extension, can we say that the integral closure of $A$ in $L$ is not Noetherian?

Answer 1

The integral closure of $\Bbb{Z}_p$ ($p$-adic integers) in $\bigcup_{n\ge 1}\Bbb{Q}_p(\zeta_{p^n-1})$ is the DVR $\bigcup_{n\ge 1}\Bbb{Z}_p[\zeta_{p^n-1}]$ (in particular Noetherian).

So there is no easy rule.